There are many important facts that go along with Bezout’s identity:

1.)
All common divisors of \(d\) are common divisors of
\(a\) and \(b\) as well

2.) As for the fact above, all divisors of \(a\) and \(b\) are also divisors of \(d\).

3.) \(a\)/\(d\) and \(b\)/\(d\) are prime integers

4.) \(a\)/\(d\)\(\left(x\right)\)+\(b\)/\(d\)\(\left(y\right)\)\(=1\)

5.) The greatest common divisor \(d\) is actually the smallest integer that can be written to satisy \(ax+by\)

6.) All integers of the form \(ax+by\) are multiples of \(d\)

Next we will work out an example.

If we have \(2x+3y=1\), well by trial and error we can quickly see that \(x\) is \(2\) while \(y\) is \(-1\). But when we are given huge numbers it can be a bit more difficult. In order to work the equation out with nigger numbers the Euclidean algorithm must be applied. Let’s recall that Euclid’s theorem states: For any numbers and \(a,b,\) \(x,\) gcd \(\left(a,b\right)\)= gcd\(\left(b,a-bx\right)\).

After having some help from the Euclidean Algorithm we can see that our equations is solvable.

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